Equation of pair of lines joining points of intersection of a line and a conic
Equation of Pair of Lines Joining Points of Intersection of a Line and a Conic
When a straight line intersects a conic section (which could be a circle, ellipse, parabola, or hyperbola), it generally does so at two points. The pair of lines that join these points of intersection can be represented by a single algebraic equation. Understanding this concept is crucial for solving problems in coordinate geometry, particularly in the context of conic sections.
Conic Sections
A conic section is a curve obtained by intersecting a cone with a plane. There are four types of conic sections:
- Circle
- Ellipse
- Parabola
- Hyperbola
Each type of conic has a standard equation:
| Conic Section | Standard Equation |
|---|---|
| Circle | $x^2 + y^2 = r^2$ |
| Ellipse | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ |
| Parabola | $y^2 = 4ax$ (or another form depending on orientation) |
| Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ |
Intersection of a Line and a Conic
A line in the plane can be represented by the equation $Ax + By + C = 0$. When this line intersects a conic section, we can find the points of intersection by solving the system of equations consisting of the line and the conic.
For example, consider the intersection of a line and a circle:
$$ \begin{align*} \text{Circle:} & \quad x^2 + y^2 = r^2 \ \text{Line:} & \quad Ax + By + C = 0 \end{align*} $$
To find the points of intersection, we would substitute $y = \frac{-Ax - C}{B}$ from the line's equation into the circle's equation and solve for $x$.
Equation of Pair of Lines
The equation representing the pair of lines joining the points of intersection can be derived using the concept of homogenization. Homogenization involves converting the equation of the conic into a homogeneous equation of the second degree by incorporating the equation of the line.
For a general conic given by $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ and a line $lx + my + n = 0$, the equation of the pair of lines joining the points of intersection is obtained by homogenizing the conic with the line:
$$ ax^2 + 2hxy + by^2 + 2gx\left(-\frac{n}{m}\right) + 2fy\left(-\frac{n}{l}\right) + c\left(\frac{lx + my}{n}\right)^2 = 0 $$
This equation is homogeneous and represents two lines passing through the origin, which are the lines joining the points of intersection when translated to the original position.
Examples
Example 1: Circle and Line
Given a circle $x^2 + y^2 = 4$ and a line $x + y - 2 = 0$, find the equation of the pair of lines joining the points of intersection.
Solution:
Homogenize the circle's equation with the line's equation: $$ x^2 + y^2 - 2(x + y) = 0 $$
This equation represents the pair of lines through the origin that, when translated, pass through the points of intersection of the original circle and line.
Example 2: Ellipse and Line
Given an ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and a line $3x + 2y - 6 = 0$, find the equation of the pair of lines joining the points of intersection.
Solution:
Homogenize the ellipse's equation with the line's equation: $$ \frac{x^2}{9} + \frac{y^2}{4} - \frac{1}{6}(3x + 2y)^2 = 0 $$
Simplify and rewrite the equation to represent the pair of lines: $$ 4x^2 + 9y^2 - (3x + 2y)^2 = 0 $$
This is the equation of the pair of lines through the origin that correspond to the lines joining the points of intersection when translated back to the original position.
Conclusion
The equation of a pair of lines joining the points of intersection of a line and a conic is a powerful tool in coordinate geometry. It allows us to represent the geometric relationship between a line and a conic section algebraically. By understanding the process of homogenization and the standard equations of conics, we can solve complex problems involving intersections and the resulting figures.